The h Bar

tbh not including that one time DS helped me on that part of the question in the problem set, I've been figuring out and answering all of this QFT stuff on my own.

0celo7

Let $(a_n)\subset\Bbb R$ be a sequence. Suppose $a_n\to x$ and $a_n\to y$. Let $\epsilon>0$. There is an $N_1\in\Bbb N$ such that $n\ge N_1$ implies $|a_n-x|<\epsilon/2$. Also, there is an $N_2\in\Bbb N$ such that $n\ge N_2$ implies $|a_n-y|<\epsilon/2$. Let $N=\max\{N_1,N_2\}$. Then for $n\ge N$, $$|x-y|\le |x-a_N|+|y-a_N|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$$ Thus $|x-y|<\epsilon$ for any $\epsilon>0$, hence $x=y$.

@BernardMeurer what?

@bloo so all that help I gave you with derivatives?

user218912

@0celo7 and that.

Bernard Meurer

$\epsilon/2$?

0celo7

oh god I broke it

Damn

That should read: Then, since $N\ge N_1$ and $N\ge N_2$, $$|x-y|\le\cdots$$

Bernard Meurer

2:38 AM

why over two?

0celo7

@BernardMeurer so when you add your two estimates together you get $\epsilon$ instead of $2\epsilon$

@BernardMeurer Maybe we should start with something simpler.

Since you can't use notes.

Bernard Meurer

Didnt I prove this over skype the other day?

0celo7

@BernardMeurer Prove: $x=y\Leftrightarrow (\forall \epsilon>0):|x-y|<\epsilon$.

Bernard Meurer

But by trichotomy

0celo7

@BernardMeurer no

that was the uniqueness of the supremum

Bernard Meurer

2:40 AM

AH

0celo7

prove what I just said pls

Bernard Meurer

Okay

0celo7

it's a good proof

very useful thing to know

Bernard Meurer

who are x and y?

0celo7

$x,y\in\Bbb R$.

Bernard Meurer

2:41 AM

$\forall x, y \in \mathbb R$

Ah

0celo7

One direction is trivial!

user218912

can't we talk about physics here

0celo7

No.

user218912

I need exposure

Bernard Meurer

Hmm

Well

Every number has a symmetric by definition

so if x=y, -y is x's symmetric

and by definiton x-x=0

since we defined eps as >0 the solution will always be true

0celo7

2:45 AM

Are you trying $\Rightarrow$?

What is "symmetric"

"solution"

Bernard Meurer

$\forall x\in \mathbb R, \exists z : x-z=0$

0celo7

But yes, that's the idea. $x=y\Rightarrow x-y=0\Rightarrow |x-y|=0\Rightarrow |x-y|<\epsilon$ for any $\epsilon>0$.

The other direction is harder

I'm going to do laundry, brb

Bernard Meurer

That is the symmetric, the notation however is that $\forall x\in \mathbb R, \text{ iff } x+z=0\text{ then } z=-x$

0celo7

You will want a contradiction

Bernard Meurer

So the other way around: we know the abs will make it positive so $|a-b| \geq 0$

Since this is for all epsilon we can suppose the lower bound case eps=1

So for that to be true $|x-y|$ need to be such that $|x-y|<1$ always

Wait

these are reals not naturals

fuck

Hm

How do I denote the infinitesimal? The smallest number larger than 0

0celo7

2:53 AM

@Slereah can tell you

but why the hell do you want it

that's not standard analysis

Bernard Meurer

@0celo7 You told me to find a solution I'm finding one

0celo7

"infinitesimal" is not a thing in real analysis

Bernard Meurer

if we assume the smallest possible value of eps we can show that the solution of x-y must be 0

Tsc, well fuck then

$$x=y\Leftrightarrow (\forall \epsilon>0):|x-y|<\epsilon$$

Dude

0celo7

yeah

Bernard Meurer

The reals are infinitely dense

0celo7

2:59 AM

that's not proper terminology

Bernard Meurer

How the fuck do I denote it then?

ffs

0celo7

denote what

Bernard Meurer

$$\forall x,y \in \mathbb R, \exists z\in \mathbb R : a<z<x$$

0celo7

The reals are a linear continuum.

Bernard Meurer

There, it's dense af

Lol, I'll write that on my exams "Since the reals are dense af we can assume..."

But okay, it's a linear continuum

0celo7

3:02 AM

No one says that in analysis though.

I think there's a special name for that property in the real number case

Bernard Meurer

If there were x,y such that the inverse implication wasn't true... Fuck I lost my train of thought with this density thing

0celo7

do you want me to tell you?

Bernard Meurer

I had it, now idk anymore

Not yet

0celo7

If I were an analysis prof I would give this on every test

even advanced analysis

I find the correct argument really tricky

it's one line

Bernard Meurer

since R is a linear continuum the only pair x,y that can exist so that the difference is smaller than every eps in R is (x,x) so then it's 0

0celo7

3:04 AM

but I always have to think about it

@BernardMeurer Proof?

Bernard Meurer

Because if it was anything else there would be infinite possible eps smaller than it

Lemme try

user116211

Hey, @0celo7, you must know the proof of rational numbers being dense in $\mathbb R\,.$

0celo7

@MAFIA36790 Uh, vaguely

Why?

user116211

While the crux of the proof is dependent on Archimedean property and I got most part of the proof, I still couldn't comprehend some thing.

0celo7

ok?

user116211

3:10 AM

Wait, ...

user116211

I didn't get why $$S = \{n\in \mathbb N| n/q\color{red}{\geq}b\}\,.$$

Bernard Meurer

$$\forall x,y \in \mathbb R, \exists z\in \mathbb R : x<z<y \\ (\forall \epsilon>0):|x-y|<\epsilon \\ x\neq y \implies |x-y| > 0 \\ \therefore \epsilon > |x-y| \implies \exists z\in \mathbb R : \text{ fuck this shit}$$

user116211

When will be the equality case in $n/q \geq b\,?$

0celo7

@BernardMeurer Want to know it?

You're close.

Bernard Meurer

3:13 AM

@0celo7 No, I want to find it

0celo7

@MAFIA36790 Reading.

user116211

@0celo7 sure.

0celo7

@MAFIA36790 How have they formulated the Archimedean property?

user116211

@0celo7 You mean how they stated the property?

0celo7

$(\forall x\in\Bbb R)(\exists n\in\Bbb N): n>x$?

@MAFIA36790 Why what?

user116211

3:14 AM

For each pair of real numbers $a$ and $b,$ there is a natural number n such that $na\gt b\,.$

0celo7

@MAFIA36790 ok same thing

user116211

@0celo7 yes.

0celo7

So what's your question?

user116211

@0celo7 In the set $S,$ why did they use the equality sign in $n/q\geq b\,?$

user116211

When will be the equality case?

0celo7

3:16 AM

when $bq$ is an integer.

i.e. if $b$ is an integer

*positive integer

user116211

$1/q$ is $b-a$ where $a\gt 0$ and $b\gt a\,.$ So, that means $b-a/q\geq b\,.$

Bernard Meurer

$$\forall x,y \in \mathbb R, \exists z\in \mathbb R : x<z<y \\ (\forall \epsilon>0):|x-y|<\epsilon \\ x\neq y \implies |x-y| > 0 \\ \therefore \epsilon > |x-y| \implies \exists z\in \mathbb R : |x-y|<z<\epsilon \\ \therefore \epsilon > 0 \implies |x-y|=0 s.t. \forall \epsilon, \epsilon > |x-y|$$

0celo7

dude

Bernard Meurer

Best I can do

0celo7

$q$ is a natural number

user116211

3:18 AM

WTH

0celo7

if $b$ is a natural number, then $bq$ is a natural number

user116211

checking

user116211

I mean unless $q=1$ and $a=0,$ how can it give the equality with $b\,?$ Also, it was stated initially that $a\gt 0\,.$

user116211

So, no chance of equality

user116211

Or am I missing something?

0celo7

3:21 AM

yes you're missing something

if $b$ is a natural number, then $bq$ is a natural number

do you agree?

user116211

@0celo7 Yes, I agree with this statement but I'm unable to see how this statement is related to the equality case.

Bernard Meurer

@0celo7 yes it's a closed algebra

0celo7

because then there is an $n\in\Bbb N$ such that $n=bq$

DanielSank

@0celo7 Solve for the amplitude of the rightward traveling wave in the rightmost region.

Bernard Meurer

@0celo7 DUDE ITS 4:30 AM

WTF

0celo7

3:23 AM

namely $bq$ itself

Bernard Meurer

@0celo7 Is my proof right even?

DanielSank

That, divided by the incoming wave amplitude, is the transmission coefficient.

user116211

@0celo7 Ahhha!!

user116211

So, is there some sort of generator that makes an element of natural number product of finite numbers of other natural numbers?

user116211

Or is it an axiom?

user116211

3:25 AM

@DanielSank o/

0celo7

@BernardMeurer no

I will tell you in one minute.

Bernard Meurer

@0celo7 I give up

I'm going to bed

See y'all

user116211

@BernardMeurer Yes, go to bed; o/

0celo7

@BernardMeurer WAIT

typing

Assume $x\ne y$. Then $|x-y|=:\epsilon_0>0$.

Then $|x-y|\not <\epsilon_0$, which contradicts $|x-y|<\epsilon$ for all $\epsilon>0$.

QED.

user116211

I got the the thing, but @0celo7, the thing you stated, is it an axiom of Natural Numbers?

0celo7

3:32 AM

@MAFIA36790 what?

what did I state?

user116211

@0celo7 Am I seeing the application of Trichotomy Law?

0celo7

@DanielSank I see.

@MAFIA36790 Yes.

user116211

10 mins ago, by 0celo7

because then there is an $n\in\Bbb N$ such that $n=bq$

0celo7

@MAFIA36790 dude

what's so hard to understand

if $bq$ is a natural number, then of course there's a natural number equal to $bq$

user116211

._.

0celo7

3:34 AM

namley $bq$ itself

user116211

@0celo7 sure.

0celo7

so there's your equality!

user116211

okkkay :)

user116211

Also, can I apply the Archimedean Property for natural numbers?

user116211

After all it was said that $a$ and $b$ were real numbers.

0celo7

3:35 AM

natural numbers are real numbers...

user116211

I can take them as natural numbers too, isn't it?

user116211

@0celo7 Yes, that's what I'm saying.

user116211

Thanks @0celo7....

Physics Meta

4:36 AM

0

Q: Can someone explain the voting system for the upcoming moderator election?

In the upcoming moderator election we each get three votes - a first, second and third choice. In order to use these effectively, I would like to understand the mechanics of how they will be used to determine the election's outcome. A side panel on the election page says We will calculate th...

DanielSank

5:03 AM

@MAFIA36790 \o

user228700

5:47 AM

Hi everyone :-)

user116211

Hello.

DanielSank

PSA: Don't migrate crap.

@Kaumudi Hi, everybody.

user116211

Stephen Hawking warns against contacting ET

user228700

@MAFIA36790 What does the term "score" mean in the language of statistics?

user116211

@Kaumudi Depends on the context of the problem.

user228700

5:50 AM

@MAFIA36790 It depends? Huh. Well, what about in the following definition:

user228700

"Measures of dispersion are descriptive statistics that describe how similar a set of scores are to each other"

DanielSank

That's really bad wording.

user228700

I understand the definition of measures of dispersion. What is the meaning of the term score here?

user116211

Very much bad.

user228700

@DanielSank Really? :/

DanielSank

5:52 AM

...unless this text is already talking about scores and you've left out context.

@Kaumudi Are you keeping context from us?

If not, then "score" is a dumb and confusing way to say "value".

user228700

@DanielSank Uh, it uses the term score multiple times before this. How does that matter? No, there is no other context to be given :/

user228700

@DanielSank Then why use the term score? Where is it used correctly, then?

DanielSank

@Kaumudi Well, then the writer has chosen to use rather non-standard vocabulary (at least, in my opinion).

user116211

They are measures which expresses the spread of observations in terms of the average of deviations from certain (central) values.

DanielSank

@Kaumudi To me, "score" is the value associated to how many times you get the ball into the goal in football.

@MAFIA36790 Correct.

user228700

5:55 AM

@DanielSank U mean like frequency?

DanielSank

@Kaumudi No.

If I kick the ball in the goal twice, and you do so three times, the score is 2-3.

user228700

Ohh.

user228700

Huh. Okay, then I'll just ignore the word for now.

DanielSank

Good idea.

Why are so many books so bad?

user228700

@DanielSank I found that on a website.

user116211

5:57 AM

@DanielSank They are not standard books, so I'll not make a comment on those books.

DanielSank

Oh, I know! Because unlike software, book sources are not made available on hosting services like github which allow people to file issues.

user116211

They are just for aggregating formulas and that's it.

DanielSank

@MAFIA36790 Not true, sir! Some books explain well. Not many though.

user116211

@DanielSank As I said, I'll not make any comment on those high-school books.

user116211

I have seen though books use score; but it's not a regular usable terminology.

user116211

6:03 AM

It varies from authors and contexts what score actually is.

user116211

Scientists discover subsurface ocean on Dione

David Z

7:21 AM

So here's a funny thing... my numerical integration code is coming up with $\int_0^1 (1 - 2\xi + 2\xi^2)\mathrm{d}\xi = 1$, when it should be $\frac{2}{3}$. If I give it $\int_0^1 \mathrm{d}\xi = 1$, it gives the right answer. If I give it $\int_0^1 (1 - 2\xi)\mathrm{d}\xi$, it gives $-\frac{2}{3}$ whereas the correct answer is $0$. If I give it $\int_0^1 (1 - \xi)\mathrm{d}\xi$ it produces $0$ (and very quickly, too) but the answer should be $\frac{1}{2}$.

Any thoughts from the hive mind on what could be wrong with my code?

oh wait, I figured it out - it was setting $\xi=1$. duh

Secret

7:48 AM

Felt proud to be able to explain current density going out of slab of thickness 2z but infinite in the xy plane, and how the B field is contributed by each J:

Caption: Consider any $z\neq 0$, then since the B field from each J decays as $\frac{1}{r^2}$ then the field contributed by each J in the slab in a given z will be asymmetric for field lines that are pointing to -y and those pointing to +y. Therefore, there will be some leftovers when all the contributions are summed together.

For $z=0$, however, the contribution is symmetric from top and bottom, thus the field always cancel there leaving no field

As for why there is no y component in the field, because the slab is infinite in the y direction, any point along any fixed y is symmetric left and right, thus loops always cancel, leaving no y component

innisfree

8:53 AM

@DavidZ why are you writing that stuff? Use a library that implements the best algorithms and has been tried, tested, debugged, profiled etc

David Z

10:09 AM

@innisfree got any suggestions?

user116211

@0celo7, You are still awake or your phantom phone is logging you still ;P

Secret

Group theory is kinda cool, especially when you start turning concepts like cosets, subgroups, torsion, generators etc. into some pictures

The idea of orbits really help in visualising most groups

innisfree

10:37 AM

@DavidZ sure, depends on the language. But generally quadpack a Fortran library, which is part of GNU scientific library (gsl) for c and cpp, and used by scipy quad in python

QUADPACK

QUADPACK is a FORTRAN 77 library for numerical integration of one-dimensional functions. It was included in the SLATEC Common Mathematical Library and is therefore in the public domain. The individual subprograms are also available on netlib. The GNU Scientific Library reimplemented the QUADPACK routines in C. SciPy provides a Python interface to QUADPACK. == Routines == The main focus of QUADPACK is on automatic integration routines in which the user inputs the problem and an absolute or relative error tolerance and the routine attempts to perform the integration with an error no larger than that...

Slereah

When will Fortran die

David Z

@innisfree I'm already using GSL

innisfree

Cool, well it's in there

gnu.org/software/gsl/manual/html_node/…

David Z

Yeah, I know, I'm quite familiar with much of GSL ;-)

innisfree

Oh sorry I misunderstand you

Uh mistunderstood you, even

The worst thing about my programming education is that we were encouraged to write numerical recipes ourselves (integration, differential equations, eigenvalues of matrices etc)

It's good to understand them, but it's crazy to make your own buggy implementations of such common things

David Z

10:47 AM

Yeah, agreed

I'm all for using libraries

Whatever my problem is, it's a bug in my "driver" code, not the integration routines themselves.

Secret

Pro-p group

In mathematics, a pro-p group (for some prime number p) is a profinite group G {\displaystyle G} such that for any open normal subgroup N ◃ G {\displaystyle N\triangleleft G} the quotient group G / N {\displaystyle G/N} is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite. Alternatively, one...

[Running gag] Slereah, I heard you like p adic numbers

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